Towards Nematic Phases in Ionic Liquid Crystals – A Simulation Study

Abstract Ionic liquid crystals (ILCs) are soft matter materials with broad liquid crystalline phases and intrinsic electric conductivity. They typically consist of a rod‐shaped mesogenic ion and a smaller spherical counter‐ion. Their mesomorphic properties can be easily tuned by exchanging the counter ion. ILCs show a strong tendency to form smectic A phases due to the segregation of ionic and the non‐ionic molecular segments. Nematic phases are therefore extremely rare in ILCs and the question of why nematic phases are so exceptional in existing ILCs, and how nematic ILCs might be obtained in the future is of vital interest for both the fundamental understanding and the potential applications of ILCs. Here, we present the result of a simulation study, which highlights the crucial role of the location of the ionic charge on the rod‐like mesogenic ions in the phase behaviour of ILCs. We find that shifting the charge from the ends towards the centre of the mesogenic ion destabilizes the liquid crystalline state and induces a change from smectic A to nematic phases.

The parameters and ′ are given by

Simulation details
The Coulomb interaction is calculated using a particle-particle-particle-mesh method [2,3] , with the accuracy set to 1 ⋅ 10 −6 . The Coulomb interaction is given by (S8) Here = 1 is a prefactor, 1 * and 2 * are reduced charges and is the distance between the two charges.
Apart from the coulomb interaction, three non-bonded interactions are defined: The Gay-Berne (GB) to GB interaction, the Lennard-Jones (LJ) to LJ interaction and the GB to LJ interaction. The parameters of the GB interaction are equal to the ones used by Beradi et al. [4] and Saielli et al. [5] . The parameters of the LJ interaction, the Coulomb interaction, the number density and the number of particles are taken from Saielli et al. [5] . The GB-to-GB interaction is modelled using the parameters 1 = 3, 2 = 5, = 1, = 3, 0 = 1, 0 = 1. The LJ-to-LJ interaction uses parameters = 1 and = 1. Here is the diameter of the LJ particle and is the potentials well depth. The LJ potential is the classic 12-6 potential. The parameters of the interaction between the GB and the LJ particles is calculated by Lorentz [6] and Berthelot [7] mixing rules [8] . It is therefore modelled like a GB interaction with parameters 1 = 2, 2 = √5, = 1, = 3, 0 = 1, 0 = 1. We note that this has one important implication for the visual display of snapshots. GB and LJ particles can look like they are overlapping in a snapshot, while they are in fact not overlapping in a physical sense (see Figure S1). Figure S1: The mixed interaction between the spherical LJ particles (green) and the prolate GB particles (yellow) is approximated accord ing to the mixing rule by the side-to-end Gay-Berne interaction of the blue ellipsoids, which leads to the blue curve GB ( ). The Coulomb interaction between the particles of opposite charge is shown by the green curve ( ), the sum of both is the total interaction potential shown by the orange curve. As can be seen, the use of the mixed interaction leads to an energy minimum which shows a partial overlap of the GB and the LJ particles, this overlap does not affect the energies as the potentials represented by the green and yellow shapes are not used for calculating the mixed interaction, the relative proportions are the same as in our simulations.

Simulation runs
Every simulation has three parts: Setup, Warmup and the actual simulation runs: During Setup the interactions are defined, the particles are randomly placed in the simulation box and the Coulomb interaction is tuned. The Coulomb interaction must be tuned using the target box size of * = 24224. After this the box size is increased by several orders of magnitude. The increase of the box size must be done, otherwise the particles might be too close together and the simulation would "explode".
Warmup consists of 200000 molecular dynamic (md) steps during which the box size is incrementally decreased until the target box size is reached. When the target box size is reached, the Coulomb interaction is retuned.
The actual simulation consists of 1000000 md steps for every temperature. The starting temperature is selected, so that the system starts from an isotropic phase. For most systems the starting temperature is * = 3.3. The temperature decrease is Δ * = 0.05 for all simulations.

Explanation of variables and colour codes
In the following tot is the total energy, kin is the total kinetic energy, nonbonded is the total energy from the interactions between the GB and GB, LJ and LJ, GB and LJ particles and coulomb is the total coulomb energy. All energies are divided by the number of particles particles . The different functions and order parameters are described in the main text of the paper. All order parameters and functions are averaged over every 1000 th snapshot from snapshot 901000-1000000 of the given reduced temperature, giving the results from 100 snapshots averaged. In snapshot pictures the LJ particles are drawn in green. The colour code of the GB particles is according to their orientation to the director . A particle with a high angle between and its long axis appears redder, while particles that have a small are more yellow. If no legend is given for directional pair correlation functions and directional density correlation functions the ones in blue correspond to the GB particles and the ones in orange correspond to the LJ particles.
System with reduced charge position * = . -overview Figure S2: Energies over reduced temperature * for the system with reduced charge position * = 0.8. Figure S3: Orientational order parameter 2 and translational order parameter Σ over reduced temperature * for the system with reduced charge position * = 0.8. Figure S4: Polar order parameter 1 over reduced temperature * for the system with reduced charge position * = 0.8. Figure S5: A picture of the 1000000 th simulation snapshot at reduced temperature * = 3.3 for the system with reduced charge position * = 0.8. The system is in the isotropic phase. Figure S6: Directional pair correlation functions at reduced temperature * = 3.3 for the system with reduced charge position * = 0.8. Figure S7: Directional density-distributions at reduced temperature * = 3.3 for the system with reduced charge position * = 0.8.
-smectic A phase Figure S8: A picture of the 1000000 th simulation snapshot at reduced temperature * = 2.95 for the system with reduced charge position * = 0.8. The system is in the smectic A phase. System with reduced charge position * = .
-overview Figure S11: Energies over reduced temperature * for the system with reduced charge position * = 0.5. Figure S12: Orientational order parameter 2 and translational order parameter Σ over reduced temperature * for the system with reduced charge position * = 0.5. Figure S13: Polar order parameter 1 over reduced temperature * for the system with reduced charge position * = 0.5.
-isotropic phase Figure S14: A picture of the 1000000 th simulation snapshot at reduced temperature * = 3.3 for the system with reduced charge position * = 0.5. The system is in the isotropic phase. Figure S15: Directional pair correlation functions at reduced temperature * = 3.3 for the system with reduced charge position * = 0.5. 8 Figure S16: Directional density-distributions at reduced temperature * = 3.3 for the system with reduced charge position * = 0.5.
-smectic A phase Figure S17: A picture of the 1000000 th simulation snapshot at reduced temperature * = 2.5 for the system with reduced charge position * = 0.5. The system is in the smectic A phase. System with reduced charge position * = .
-overview Figure S20: Energies over reduced temperature * for the system with reduced charge position * = 0.4375. Figure S21: Orientational order parameter 2 and translational order parameter Σ over reduced temperature * for the system with reduced charge position * = 0.4375. System with reduced charge position * = . at * = .
-isotropic phase Figure S23: A picture of the 1000000 th simulation snapshot at reduced temperature * = 2.8 for the system with reduced charge position * = 0.4375. The system is in the isotropic phase.  System with reduced charge position * = . at * = .
-smectic A phase System with reduced charge position * = .
-overview Figure S32: Energies over reduced temperature * for the system with reduced charge position * = 0.375. Figure S33: Orientational order parameter 2 and translational order parameter Σ over reduced temperature * for the system with reduced charge position * = 0.375. Figure S34: Polar order parameter 1 over reduced temperature * for the system with reduced charge position * = 0.375.
-isotropic phase Figure S35: A picture of the 1000000 th simulation snapshot at reduced temperature * = 3.3 for the system with reduced charge position * = 0.375. The system is in the isotropic phase. System with reduced charge position * = . at * = .
-smectic A phase System with reduced charge position * = .
-overview Figure S44: Energies over reduced temperature * for the system with reduced charge position * = 0.3125. Figure S45: Orientational order parameter 2 and translational order parameter Σ over reduced temperature * for the system with reduced charge position * = 0.3125. Figure S46: Polar order parameter 1 over reduced temperature * for the system with reduced charge position * = 0.3125.
-isotropic phase Figure S47: A picture of the 1000000 th simulation snapshot at reduced temperature * = 2.8 for the system with reduced charge position * = 0.3125. The system is in the isotropic phase. Figure S48: Directional pair correlation functions at reduced temperature * = 2.8 for the system with reduced charge position * = 0.3125. System with reduced charge position * = . at * = .
-smectic A phase System with reduced charge position * = .
-overview Figure S56: Energies over reduced temperature * for the system with reduced charge position * = 0.25. Figure S57: Orientational order parameter 2 and translational order parameter Σ over reduced temperature * for the system with reduced charge position * = 0.25. Figure S58: Polar order parameter 1 over reduced temperature * for the system with reduced charge position * = 0.25.
-isotropic phase Figure S59: A picture of the 1000000 th simulation snapshot at reduced temperature * = 3.3 for the system with reduced charge position * = 0.25. The system is in the isotropic phase. Figure S60: Directional pair correlation functions at reduced temperature * = 3.3 for the system with reduced charge position * = 0.25. Figure S61: Directional density-distributions at reduced temperature * = 3.3 for the system with reduced charge position * = 0.25.
-nematic phase Figure S62: A picture of the 1000000 th simulation snapshot at reduced temperature * = 1.8 for the system with reduced charge position * = 0.25. The system is in the nematic phase. System with reduced charge position * = . at * = .
-crystalline phase System with reduced charge position * = .
-overview Figure S71: Energies over reduced temperature * for the system with reduced charge position * = 0.125. Figure S72: Orientational order parameter 2 and translational order parameter Σ over reduced temperature * for the system with reduced charge position * = 0.125. Figure S73: Polar order parameter 1 over reduced temperature * for the system with reduced charge position * = 0.125.
-isotropic phase Figure S74: A picture of the 1000000 th simulation snapshot at reduced temperature * = 3.3 for the system with reduced charge position * = 0.125. The system is in the isotropic phase.  System with reduced charge position * = . at * = .
-nematic phase System with reduced charge position * = .
-overview Figure S83: Energies over reduced temperature * for the system with reduced charge position * = 0.0. Figure S84: Orientational order parameter 2 and translational order parameter Σ over reduced temperature * for the system with reduced charge position * = 0.0. The system transitions into a crystalline phase starting at * = 0.9 and at this point the system has free volume. Due to the free volume the calculation of the translational order parameter is unreliable for temperatures lower than * = 0.9. Figure S85: Polar order parameter 1 over reduced temperature * for the system with reduced charge position * = 0.0. Figure S86: A picture of the 1000000 th simulation snapshot at reduced temperature * = 2.5 for the system with reduced charge position * = 0.0. The system is in the isotropic phase. Figure S87: Directional pair correlation functions at reduced temperature * = 2.5 for the system with reduced charge position * = 0.0. System with reduced charge position * = . at * = .
-crystalline phase Figure S92: A picture of the 1000000 th simulation snapshot at reduced temperature * = 0.9 for the system with reduced charge position * = 0.0. The system is in a crystalline phase. a) b) Figure S93: Directional pair correlation functions at reduced temperature * = 0.9 for the system with reduced charge position * = 0.0. Calculated parallel (a) and orthogonal (b) to the director of the GB particles. a) b) Figure S94: Directional density-distributions at reduced temperature * = 0.9 for the system with reduced charge position * = 0.0. Calculated parallel (a) and orthogonal (b) to the director of the GB particles.